University of Glasgow Department of Physics and Astronomy Problems Handbook 2006 2007 Notation: Marks are shown in square brackets in the right-hand margin. Question numbers followed by a small e are recent exam questions. Answers to some numerical problems are shown in curly brackets. You may need to use some of the following constants in your solutions to the problems: Values of constants speed of light c 2.998 10 8 m s 1 gravitational constant G 6.673 10 11 N m 2 kg 2 Planck constant h 6.626 10 34 J s Boltzmann constant k 1.381 10 23 J K 1 Stefan-Boltzmann constant σ 5.671 10 8 W m 2 K 4 Rydberg constant R 1.097 10 7 m 1 Avogadro constant N A 6.022 10 23 mol 1 gas constant R 8.315 J mol 1 K 1 proton mass m p 1.673 10 27 kg electron mass m e 9.109 10 31 kg elementary charge e 1.602 10 19 C electronvolt ev 1.602 10 19 J astronomical unit AU 1.496 10 11 m parsec pc 3.086 10 16 m light year ly 9.461 10 15 m solar mass M 1.989 10 30 kg solar radius R 6.960 10 8 m solar luminosity L 3.826 10 26 W Earth mass M 5.976 10 24 kg Earth radius R 6.378 10 6 m obliquity of the ecliptic ɛ 23 26
CONTENTS 1 Contents 1 General Astronomy 2 2 Positional Astronomy 4 3 Dynamical Astronomy 10 4 Solar System Physics 16 5 Stellar Motion 29 6 Observational methods 30 7 Stellar Astrophysics 36 8 Compact objects 42 9 Cosmology 48
2 A1X/Y Problems, 2006 2007 1 General Astronomy A1X General Astronomy 1.1e Describe a method of determining the radius of a planetary orbit (or semi-major axis) (a) as a multiple of the Earth s orbital radius (b) absolutely i.e., in metres. [5] 1.2e Explain with the aid of orbital diagrams how the loop-like motions of an outer planet as seen against the stellar background can be explained using (a) the Ptolemaic system (b) the heliocentric system. [5] 1.3 Define the term Astronomical Unit (AU). [3] The maximum angular elongation of Venus from the Sun, seen from Earth, is η max = 46. Find the radius of Venus s orbit (assumed circular) in AU. {0.719 AU} [4] If at this configuration a radar signal takes 11 m 35 s for the Earth-Venus round trip, find the value of 1AU in metres. (Neglect the motion of Venus and the Earth during the time of the radar signal round-trip.) {1.499 10 11 m} [5] Explain, with a diagram, why successive conditions of maximum elongation do not provide the same measured values of η max. [3] Apply Kepler s 3rd Law of Planetary Orbits (a 3 P 2 ) to find the shortest radio communication time from the Earth to the Voyager probe when at Neptune (orbital period 164.3 yr). {4 h 1 m 30 s } [5] 1.4 The orbital periods of Venus, Jupiter and Saturn around the Sun are given in the table below. Calculate their mean distances (in AU) from the Sun. Orbital Minimum Distance Maximum Physical Maximum period angular from angular diameter distance from diameter Sun diameter Earth Venus 225 days 10 Jupiter 12 years 30 Saturn 30 years 15 Explain why the angular diameters of the planets vary with their phase. The minimum angular size of Venus is 10. At what phase does this take place? What is the appropriate maximum angular size of Venus? Calculate the physical diameter of Venus. The table gives the approximate minimum angular sizes of Jupiter and Saturn. What are their respective maximum angular sizes? Calculate their physical diameters. Estimate the flux of the reflected light to arrive at the Earth from each of the planets when it is furthest away from the Earth as a fraction of the direct flux of the Sun. (Consider both inferior and superior planets.)
1 General Astronomy 3 PSfrag replacements Planet rsp Sun r PE r SE Earth The differences in apparent magnitude of two celestial bodies is given in terms of their respective fluxes by m 1 m 2 = 2.5 log F 1 F 2 Estimate the difference in apparent magnitude of Jupiter at maximum and minimum brightness. Discuss: When do you expect Venus to be brightest? Other stars may have planetary systems. Would you expect the occultation of the star by a planet to produce a change in the star s apparent magnitude? Write down an expression for the change in magnitude, and sketch how this would vary with time. Would such a change be measurable? How might you distinguish the variation in apparent magnitude due to occultation from the variation produced by possible star spots? 1.5e On occasion, Eros, a minor planet, approaches the Earth to within a distance of 30 10 6 km. It is suggested that its distance might be measured absolutely by parallax using two observatories separated by 1 000 km. If the accuracy of positional measurement is ±0.001 arcsec, estimate the uncertainty in the distance determination. [5] [1 radian = 206 265 arcsec] A1Y General Astronomy 1.6e Two stars radiate as black bodies with the maximum brightness at wavelengths, λ max, of 500 nm and 700 nm. It is also known that the first star has a radius three times larger than the second. What is ratio of their luminosities? [5] 1.7 A visual double star has a measured parallax of 0.105 arcsec and the semi-major axis of the orbit subtends 21.47 arcsec. If the orbital period is 350 years, calculate the sum of the masses of the stars. {69.79M } From measurements of the system s proper motion, it was found that at one epoch the subtended angles from the centre of gravity were 3.84 and 7.12 arcsec. Determine the individual masses of the stars. {45.3M ; 24.49M } [12] 1.8 (a) A star of apparent magnitude 6 shows an annual parallax of 0.02. What is its distance in parsecs, AU and metres? {50 pc; 1.03 10 7 AU; 1.55 10 18 m} [8]
4 A1X/Y Problems, 2006 2007 (b) A second star, with an identical spectrum, has apparent magnitude 11. Find its distance in parsecs. { 500 pc} (c) Find the absolute magnitude of each star. {2.5} 1.9 The hydrogen-alpha spectrum line of hydrogen (λ 0 = 656.3 nm) is observed from two galaxies. In the first it is seen at λ = 694.2 nm and in the second at λ = 765.1 nm. Find the ratio of the distances of these two galaxies, using Hubble s law. Taking values of H and of c from the front cover, find the distances in Mpc, and as fractions of the (Hubble) radius of the visible Universe. {2.87; 0.0577; 0.166} 1.10 (a) Two stars provide a flux ratio of 100:1. What is their apparent difference of magnitude? {5} (b) A solar-type star has an absolute magnitude of 4.8. It appears in a catalogue as having an apparent magnitude of 9.6. What is its distance? {91.3 parsec} 1.11 An X-ray source has a luminosity at 2 kev of 10 26 W. Convert 2 kev into Joules and hence calculate the photon production rate ( number of photons per second) emitted by this source at this energy. 1.12 A radio telescope is tuned to operate at 3 000 MHz. Calculate (a) the wavelength of the received radiation (b) the energy in ev of the photons at this frequency. 1.13 A star is measured to have a parallax of 0.045 arcsec. Calculate its distance in parsecs, metres and light years. 2 Positional Astronomy 2.1e Draw a celestial sphere for north latitude 45, indicating clearly the zenith point, the north celestial pole and the horizon. Mark in the cardinal points of the horizon and draw in the celestial equator. The date is March 21; mark in the Sun s position at noon. [5] 2.2e List Galileo s principal astronomical observational discoveries. Explain which of these could be used as evidence to confirm the heliocentric theory of Copernicus. [5] 2.3e Draw a celestial sphere to indicate clearly the relationship between the equatorial coordinate system of right ascension and declination and the coordinate system of ecliptic latitude and longitude. Indicate the Sun s position on 21 June. What is its declination at this time? [5] 2.4e Draw a celestial sphere for an observer on the Earth s equator. Mark in the north and south celestial poles and the celestial equator. Mark in the Sun s approximate position at noon on 21st December. [5]
2 Positional Astronomy 5 2.5e Indicate clearly on a diagram the position of an inferior planet with respect to the Earth and the Sun for the following configurations: (a) Inferior conjunction (b) Superior conjunction (c) Maximum elongation east (d) Maximum elongation west. [5] 2.6e Define a great circle on the surface of a sphere. Explain why this type of curve is particularly important. What is the length of the great circle arc joining the two points on the surface of the Earth with geographical latitude and longitude (60 N, 90 E) and (60 N, 90 W)? [5] 2.7 Two places A and B on the same parallel of latitude 38 33 N are 123 19 apart in longitude. Calculate, in nautical miles, (a) their distance apart along the parallel (b) the great circle distance AB. {5786.5 nmi; 5219.7 nmi} 2.8e Define carefully the coordinate system of terrestrial latitude and longitude and compare it with the celestial coordinate system of right ascension and declination. [6] Derive a formula to give the shortest distance between two points on the same parallel of latitude φ which differ in longitude by an amount λ. Explain carefully how this distance may be expressed in nautical miles. [5] A star is observed at the zenith at Glasgow University Observatory (55 54 N, 4 18 W). At the same instant of time a second star is at the zenith at the University of Helsinki Observatory (60 09 N, 24 57 E). Calculate the right ascension and the declination of each star if the Greenwich sidereal time at this instant is 16 h 41 m. [6] 2.9e Define carefully the terms conjunction, opposition and quadrature as they are applied to a superior planet. Define further the sidereal period and the synodic period for such a planet and explain how they are related. [5] The planet Mars moves round the Sun in an orbit of semi-major axis 1.52 astronomical units. Calculate its sidereal period of revolution in years. [2] Making the approximation that the orbits of Mars and the Earth are circular and coplanar, calculate the synodic period of Mars in years and the interval of time in days between opposition and quadrature for this planet. [7] Mars is observed at quadrature on March 21. Calculate its right ascension and declination at this time. [3] (Obliquity of the ecliptic ɛ = 23 26 ) 2.10e Define carefully what is meant by a parallel of latitude and a meridian of longitude, illustrating your definition with a diagram. Explain the conventions that are used in assigning a latitude
6 A1X/Y Problems, 2006 2007 and a longitude to a position on the surface of the Earth. Give a formal definition of the nautical mile. [6] State, without proof, the cosine formula of spherical trigonometry. [2] Prove that the shortest distance between the two points P 1 and P 2 on the Earth s surface with geographical coordinates (φ 1, λ 1 ) and (φ 2, λ 2 ), latitude and longitude respectively, is given by cos d = sin φ 1 sin φ 2 + cos φ 1 cos φ 2 cos(λ 1 λ 2 ). Find the length in nautical miles of the shortest air route from Dunedin (45 51 S, 170 30 E) to Los Angeles (33 51 N, 118 21 W). [5] 2.11e Define carefully the coordinate system of ecliptic longitude and ecliptic latitude. Indicate how a star s ecliptic coordinates (λ, β) are related to its right ascension and declination (α, δ) by identifying the appropriate spherical triangle on the celestial sphere. In particular prove that [4] cos λ cos β = cos α cos δ. [8] Estimate the date when the Sun s ecliptic longitude is 45 and calculate the Sun s right ascension and declination for that date. [12] (Obliquity of the ecliptic, ɛ = 23.5 ) 2.12 An aircraft leaves Lima (12 10 S, 77 05 W) and flies directly to Rome (41 53 N, 12 33 E). Draw a diagram marking the position of Lima and Rome, and the great circle joining them. (a) Calculate the distance travelled in nautical miles. (b) Calculate the aircraft s bearing as it approaches Rome. (c) Determine the longitude of the point on the flight where the aircraft crosses the Equator. {5869 nmi, 99.357 69, 63 35 } 2.13 Prove that, in an equilateral spherical triangle, the sides and angles satisfy the condition that sec A sec a = 1. [20] 2.14 Two seaports are on the same parallel of latitude 42 27 N. Their difference in longitude is 137 36. Ship A sails along the parallel of latitude from one port to the other, while ship B follows the most direct great circle route. Ship B sails at a constant speed of 20 knots. How long will it take to complete the voyage? {10 d 20 h 48 m } [12] What average speed will ship A have to maintain to complete its voyage in the same time as ship B? {23.36 knots} 2.15 Calculate the length in degrees of the great circle arc joining San Francisco (37 40 N, 122 25 W) and Tokyo (35 48 N, 139 45 E). {76.95 } [9] What is the shortest distance between these two cities in nautical miles? {4617.03 nmi} [2] An aircraft leaves San Francisco at 10 pm local time on August 21 and flies directly to Tokyo maintaining an average speed of 480 knots. Calculate [8]
2 Positional Astronomy 7 (a) the duration of the flight {9 h 18 m } [3] (b) the local time and date of arrival in Tokyo. {00:18} [6] 2.16 Using the result of the previous question, determine the direction in which an aircraft should depart from San Francisco to follow the most direct route to Tokyo. Calculate further the latitude and longitude of the most northerly point that will be reached on this direct route. {56 34 W of N; 48 39 N; 169 38 W} 2.17 The altitudes of a star at upper and lower transits (both north of the zenith) are 65 23 and 14 01. Find the latitude of the observer and the star s declination and calculate the star s altitude, azimuth and hour angle when it is at its maximum azimuth west. {δ = 64 19 ; φ = 39 42 ; a = 45 08 ; A = 37 17 } 2.18 Calculate the azimuth of the Sun at rising on Midsummer s day at Stonehenge (latitude 50 10 N) at a time when the obliquity of the ecliptic was 23 48. 2.19 Describe the celestial sphere and the diurnal motions of the stars as they would appear for an observer at (a) the Earth s North Pole (b) a point on the Earth s equator. 2.20 Prove that the maximum azimuth (east or west of north) of a circumpolar star is A = sin 1 (cos δ sec φ). The right-hand side of this equation is undefined when δ < φ. How do you account for this? 2.21 Define the phase angle, φ, and the elongation, η, of a planet illustrating your definition with diagrams for an inferior and a superior planet. [5] Why is the phase angle so called? [3] Assuming that the Moon is at a distance of 3.84 10 5 km from the Earth, calculate its elongation to the nearest arcminute when it is observed to be exactly half illuminated.89 51 [12] 2.22 An asteroid is moving in an orbit of semi-major axis 2.87 astronomical units. Calculate its sidereal period of revolution. {4.86 yr} [3] Assuming that the asteroid is moving in a circular orbit in the ecliptic plane, calculate the synodic period in years. {1.26 yr} [3] What is the maximum phase angle and the minimum phase for this asteroid? {20.39 ; 0.969} [5] Work out the interval of time in days between opposition and the occurrence of this minimum phase. {0.24 yr} [9]
8 A1X/Y Problems, 2006 2007 2.23 Draw a celestial sphere for an observer at latitude 56 N indicating the horizon and the celestial equator. Mark in the north, south and west points of the horizon and also the north celestial pole and the zenith point. [6] A star has declination 18. Indicate its approximate position on the diagram when it is setting and calculate its azimuth at that instant. {303 33 } [7] What is the hour angle of the star when it sets? {7 h 55 m } [5] How long is the star above the horizon in (a) sidereal time? {15 h 50 m } [1] (b) solar time? {15 h 48 m } [1] 2.24 (a) What is the ecliptic latitude and longitude of the Sun on May 1st? (b) Calculate the Sun s right ascension and declination for this date, assuming that the obliquity of the ecliptic ɛ = 23.5. (c) Calculate the hour angle of the Sun at sunset for this date for an observer at Glasgow, latitude 56 N, longitude 4 15 W. (d) What is the interval (in solar time) between sunrise and sunset for this observer on this date? Determine the Greenwich mean times of sunset and sunrise. (e) What is the local sidereal time and what is the Greenwich sidereal time of sunset for the Glasgow observer on May 1st? 2.25 The planet Jupiter is moving in an orbit of semimajor axis 5.202 astronomical units. Calculate its sidereal period in years. Calculate the synodic period of Jupiter in days making the approximation that both Jupiter and the Earth move round the sun in circular coplanar orbits. Calculate the interval of time in days during which Jupiter s sidereal motion will be retrograde. 2.26e Draw a celestial sphere for an observer on the Earth s equator. Indicate the positions of the north and south celestial poles and draw in the Sun s diurnal path for June 21 and December 21. [5] 2.27e Show that the sidereal motion of a superior planet must be retrograde at opposition and direct at quadrature. [5] 2.28e Draw a geocentric celestial sphere, marking clearly the celestial equator and ecliptic. State the Sun s right ascension, declination, ecliptic longitude and ecliptic latitude for each of the following dates: [1] (a) March 21 [1] (b) June 21 [1] (c) September 21 [1] (d) December 21. [1]
2 Positional Astronomy 9 2.29e Explain carefully how the latitude and longitude of a point on the Earth s surface are defined. [4] Two points on the Earth s surface have latitude and longitude (φ 1, λ 1 ) and (φ 2, λ 2 ) respectively. Prove that the shortest distance between them is d, where cos d = sin φ 1 sin φ 2 + cos φ 1 cos φ 2 cos (λ 1 λ 2 ). How would you express this distance in nautical miles? [5] On March 21 the planet Mercury had right ascension and declination (0 h 45 m, 5 04 ). Calculate its elongation from the Sun at that time. [8] 2.30e Define precisely what is meant by a circumpolar star. Show that the condition for a star to be circumpolar is that its declination δ satisfies the inequality δ > 90 φ where φ is the latitude of the observing site. Show further that if φ > 45 than all stars that transit north of the zenith are circumpolar. [5] Show that if the Sun becomes circumpolar for part of the year for all points within the arctic circle, i.e. φ > (90 ɛ), where ɛ is the obliquity of the ecliptic. Show further that this period of continuous daylight (midnight Sun) persists while the ecliptic longitude λ of the Sun satisfies the equation sin λ > cos φ cosec ɛ [6] Calculate the length of the period of continuous daylight for an observer at latitude 72, on the assumption that the ecliptic longitude of the Sun increases strictly uniformly during the year. [6] [Obliquity of the ecliptic ɛ = 23 26.] 2.31 Define what is meant by a circumpolar star using a diagram to illustrate your definition. [4] The altitude of such a star is 75 23 at upper transit and 18 09 at lower transit. Both of these transits occur north of the zenith. Find the latitude of the observing site and the declination of the star. [7] Calculate the maximum azimuth that the star can have and its altitude and hour angle at that time. [9] 2.32 The Sun is observed to set at 8.00 pm (British Summer Time) at the location of Glasgow (55 52 N, 4 15 W). (a) What is the Sun s hour angle at this time? (b) Calculate the Sun s declination. (c) On the assumption that the obliquity of the ecliptic is 23 26, calculate the Sun s ecliptic longitude at this time. (d) What are the two possible dates for this observation?
10 A1X/Y Problems, 2006 2007 2.33 In each answer, draw a suitably labelled diagram showing the positions of the transits, the celestial pole and the zenith. (a) An observer at latitude 48 0 N observes a star of declination δ = 60 0 N. Prove that this star is circumpolar, and calculate its altitude at upper and lower transit. [8] (b) Prove that for a circumpolar star of declination δ to have its upper transit south of the zenith for an observer at latitude φ, φ > δ. [5] (c) A star has zenith distances at lower and upper transits of 24 N of zenith, and 74 S of zenith, respectively. Calculate φ and δ. [7] 2.34 Show that the number of hours of daylight on a given day can be estimated as 2 cos 1 ( tan δ tan φ ). where δ is the solar declination, and φ is the observer s latitude. Calculate the hour angle of the setting of the sun for an observer at latitude 54 55 N, given that the sun s ecliptic longitude is 49 49. What date does this correspond to? 2.35 The star γ Dra has declination 51 29 27". When observed from Strasbourg, it s azimuth ranges between 289 44 32" and 70 15 28". What is the latitude of the observatory at Strasbourg? 3 Dynamical Astronomy 3.1e State Newton s law of gravitation, defining all the symbols you use. Hence derive an expression for the surface gravity of a spherically symmetrical uniform planet in terms of its mass, radius, and the gravitational constant, G. Calculate the value of the surface gravity of Jupiter, which has a mass of 1.9 10 27 kg and a radius of 7.14 10 7 m. [5] 3.2e Show, assuming Newton s law of gravitation, that the gravitational potential energy, V (r), of a small body of mass m at a distance r from a planet of mass M is given by V (r) = GmM, r where G is the constant of gravitation. [5] 3.3e State Kepler s Laws of Planetary Motion. Define the term Angular Momentum and explain, using the case of a circular orbit, the significance of Kepler s Second Law in terms of angular momentum. [5] 3.4e A body of mass m moves under gravity in a circular orbit of radius R about a body of much larger mass M. Given that the centripetal force on a body of mass m moving with velocity v in a circular orbit of radius R is mv 2 /R show that the period of the orbit is given by the equation where G is the constant of gravitation. T 2 = 4π 2 R 3 GM, [5]
3 Dynamical Astronomy 11 3.5 State Newton s law of gravitation. Demonstrate, for a circular orbit, how it is consistent with Kepler s second and third laws of planetary motion. [9] Show that the orbital period of a mass in a circular orbit of radius r about a much larger mass M is T 2 r = 4π 2 3 GM. What is the corresponding expression for an elliptical orbit? [5] The comet Faye is in an orbit with a period of 7.38 years, an eccentricity of 0.576 and a perihelion distance of 1.608 AU. The comet Wolf has a period of 8.43 years, and a perihelion distance of 2.507 AU. Calculate the eccentricity of its orbit. {0.395} [6] 3.6e A body of negligible mass m is in an elliptical orbit around the Sun. Write down an expression for the total energy, E, of the body and show that when it is at a distance r from the Sun its velocity, v, can be written as ( ) 2 v 2 = GM r + C, where the constant C = 2E/(G Mm) and G is the gravitational constant. [6] Given that the perihelion and aphelion distances of the orbit are a(1 e) and a(1 + e), where a is the semi-major axis and e is the eccentricity of the orbit, use the above expression to find the ratio of the velocity of the body at perihelion to that at aphelion. [2] From conservation of angular momentum derive a second expression for this ratio of velocities. [2] Hence show that v 2 = GM ( 2 r 1 ). a The comet Wolf has an eccentricity of 0.395 and a perihelion distance of 2.507 AU. Calculate the velocity of the comet at perihelion. [4] [3] 3.7e If a body of negligible mass is in orbit around a mass M, its speed, v, is given by v 2 = GM ( 2 r 1 ), a where r is the distance between the centres of the two masses, a is the semi-major axis of the orbit and G is the gravitational constant. An artificial satellite is in a circular orbit about the Earth at a height of 1 000 km above the surface. Use the above expression to calculate the speed of the satellite. [3] The satellite is to be transferred to an orbit of radius 30 000 km. Describe an economical way of carrying out this transfer and calculate the change in speed required to inject it into its transfer orbit. [6] (GM for the Earth = 3.989 10 14 m 3 s 2. Radius of Earth = 6 378 km.)
12 A1X/Y Problems, 2006 2007 3.8e Explain what is meant by the escape velocity from the surface of a planet. Given that the gravitational potential energy, V, of a mass m at the surface of a planet of mass M and radius R is V = Gm M/R, derive an expression for its escape velocity from the planet s surface. [4] The orbit of the Moon around the Earth has a semi-major axis of 384 400 km and a period of 27.32 days. Given that the radius of the Earth is 6 378 km, calculate the acceleration due to gravity and the escape velocity at the Earth s surface. [8] 3.9 Derive an expression for the period, T, of a body of negligible mass in a circular orbit with radius R about the Earth. [9] Derive the height above the equator for a geostationary communications satellite, given that the radius of the Earth is 6.38 10 6 m and the (surface) acceleration due to gravity is 9.8 m s 2. [8] How many satellites are needed to give complete coverage of the Earth (excluding polar regions)? [2] 3.10 State Newton s Law of Gravitation and show that the acceleration due to gravity at the surface of a planet of mass M and radius R is given by g = GM R 2, where G is the gravitational constant. [3] Given that the mass of Venus is 0.817 times that of the Earth and the radius of Venus is 0.97 times that of the Earth, calculate the acceleration due to gravity at the surface of Venus if its value at the surface of the Earth is 9.8 m s 2. {8.5 m s 2 } [3] Show that the gravitational potential energy of a mass, m, a distance r from the centre of a planet of mass M ( m) is V (r) = GMm, r and write down an expression for the total energy. { 1 2 mv2 G Mm/r} [8] Explain the meaning of the phrase escape velocity from the surface of a planet and derive an expression for it in terms of the radius of the planet and the acceleration due to gravity at its surface. Given that the radius of the Earth is 6.38 10 6 m and using the data given and derived earlier calculate the escape velocity from the surface of Venus. {10.3 km s 1 } [6] 3.11 By conservation of energy and angular momentum for a small body in an orbit about the Sun we can get the relations E = 1 2 mv2 GMm, and h = rv sin θ, r where M and m are the masses of the Sun and the body respectively, r is the distance of the body from the Sun, G is the gravitational constant, v is the speed of the body, θ is the angle its direction of motion makes with a line to the body from the Sun, and h is a constant. Explain the origin of the terms in the above equations. [3]
3 Dynamical Astronomy 13 If the orbit is an ellipse of semi-major axis a and eccentricity e, show (by considering the perihelion and aphelion of the orbit) that ( 2 v 2 = GM r 1 ) a and h 2 = GMa(1 e 2 ). [10] An asteroid is detected at a distance of 1.26 AU from the Sun. Its velocity is 33.3 km s 1 with directed at 74.7 to the line joining the two. Calculate the semi-major axis and period of its orbit. Does the orbit cross that of the Earth? [7] (Use Solar units for this problem, remembering that 1 km s 1 = 0.211 AU yr 1.) 3.12 Show that the laboratory value of the gravitational constant can, with other measurements given below, give the masses of the Sun and the Earth. semi-major axis of Earth s orbit radius of Earth surface gravitational acceleration on Earth Earth year 1.496 10 8 km 6371 km 9.81 m s 2 365.25 d 3.13 Assuming the orbit of Mars to be circular and in the ecliptic with a synodic period of 780 days calculate (without using Kepler s third law): (a) the orbital period in days (b) the radius of the orbit in astronomical units, given that, 35.61 days after opposition, the elongation is 139 05 (c) the interval between opposition and the next quadrature. 3.14 The satellite Europa describes an orbit about Jupiter of semi-major axis 6.71 10 5 km in a period of 3.552 days. Neptune s satellite Triton has semi-major axis 3.55 10 5 km and orbital period 5.877 days. Calculate the ratio of the masses of the two planets. {18.49} 3.15 Halley s comet has an orbital period of 76 years and its perihelion distance is 0.59 AU. Calculate its semi-major axis and its greatest distance from the Sun in AU. Use your lecture notes to calculate its velocity at perihelion and aphelion in AU yr 1. Convert these to km s 1, given that the Earth s orbital velocity is 29.8 km s 1. What is the ratio of its greatest and least orbital velocities? {17.942 AU; 35.294 AU; 11.47 AU yr 1 ; 0.192 AU yr 1 ; 59.7} 3.16 A comet, moving towards perihelion, has a velocity of 31.53 km s 1 when 1.70 AU from the Sun, directed at 143.16 from the direction of the radius vector. Use this information to calculate a and h. Also obtain the eccentricity, perihelion and aphelion distances and period of the orbit. {a = 17.84 AU; e = 0.966; 0.607 AU; 35.07 AU; 75.35 yr} 3.17 The satellite Phobos of Mars has an orbit with a period of 0.318 9 days and a semi-major axis of 9.38 10 3 km. The diameter of Mars is 6 762 km. Show that, without a knowledge of the gravitational constant, this is sufficient to calculate the surface gravity of Mars and its surface escape velocity. Obtain values for these quantities.
14 A1X/Y Problems, 2006 2007 3.18 Explain what is meant by a Hohmann transfer orbit. A manned spacecraft leaves the vicinity of the Earth to go to Venus using the most economical transfer orbit. Calculate the minimum change in velocity required to reach the orbit of Venus and the time taken for the transfer. If the spacecraft is to rendezvous with Venus, what should be the position of Venus relative to the Earth in order to achieve this? Calculate the change in velocity required once the spacecraft reaches Venus. How long is required before the spacecraft can return to Earth using a Hohmann transfer orbit? Assume that the orbits of Earth and Venus are circular and coplanar with radii of 1.0 and 0.723 AU respectively. Ignore the gravitational effects on the spacecraft by Earth and Venus. 3.19 The semi-major axis of the orbit of Mars is 1.524 AU and the orbital eccentricity is 0.093. Assuming the Earth s orbit to be circular and coplanar with that of Mars, calculate: (a) the distance of Mars from the Earth at closest approach (b) the ratio of the speeds of Mars in its orbit at perihelion and aphelion (c) the speed of Mars at aphelion in AU per year. 3.20 Two artificial satellites are in elliptical orbits about the Earth and both have the same period. The ratio of the velocities at perigee is 1.5 and the eccentricity of the satellite with the greater perigee velocity is 0.5. Calculate the eccentricity of the orbit of the other satellite and the ratio of the apogee velocities of the two satellites. 3.21 Halley s Comet moves in an elliptical orbit with an eccentricity of 0.967 3. Calculate the ratios of (a) the linear velocities (b) the angular velocities at aphelion and perihelion. 3.22 The period of Jupiter is 11.86 years and the masses of Jupiter and the Sun are respectively 3.3 10 5 and 318 times that of the Earth. Calculate the change in Jupiter s orbital period if the semi-major axis was the same but its mass was the same as the Earth. 3.23 A lunar probe is put into an elliptical transfer orbit from a circular parking orbit (radius 6 878 km) about the Earth. It is intended that the apogee of the transfer orbit should touch the Moon s orbit (assumed circular with a radius of 3.844 10 5 km). If the velocity in the parking orbit is 7.613 km s 1, calculate: (a) the semi-major axis and eccentricity of the transfer orbit (b) the time the probe takes to reach apogee (c) the required velocity increment to give the transfer orbit. 3.24 Explain what is meant by the linear momentum of a body and describe how it changes when a force is applied to the body. (a) Explain what is meant by the angular momentum of a body about an axis. What quantity (analogous to force) changes the angular momentum of a body?
3 Dynamical Astronomy 15 (b) Write down Kepler s laws of planetary motion. Explain, using a circular orbit, the physical basis of the second law. (c) Show how a cone can be sliced to give a circle, an ellipse or a hyperbola. Where does a parabola fit in here? (d) Consider a satellite in a circular orbit about the Earth. At a particular point in the orbit the spacecraft engines are fired to increase the tangential velocity of the craft. Sketch the possible trajectories of the spacecraft. (e) Sketch an ellipse and mark clearly the positions of the foci. If the Sun is at one focus of an elliptical orbit, mark the positions of perihelion and aphelion and write down expressions for the perihelion and aphelion distances in terms of the semi-major axis and eccentricity of the ellipse. The comet Encke has an orbit with a period of 3.3 years, a perihelion distance of 0.339 AU and an eccentricity of 0.847. Comet Halley has an orbit with perihelion distance of 0.587 AU and eccentricity of 0.967 AU. Calculate the length of the semi-major axis for each of the above orbits and using Kepler s third law find the period of comet Halley. 3.25e State Kepler s three laws of planetary motion. Given that the semi-major axis of the orbit of Venus is 0.723 3 AU, calculate its sidereal period in years. [5] 3.26e The orbital period, T, of a body of mass m in an elliptic orbit about another body of mass M is [ a 3 ] 1/2 ( a 3 ) 1/2 T = 2π = 2π, G(M + m) µ where a is the orbital semi-major axis and G the gravitaional constant. Also if v is the speed of the body of mass m when its radius vector is r, then ( 2 v 2 = µ r 1 ). a Show that, if at any time the direction of motion of m is changed without changing the magnitude of the velocity and without changing the length of the radius vector r, the semimajor axis a and the period T of the resulting orbit remain unaltered. [5] 3.27e A planet of mass m is in orbit about the Sun, of mass M. The sum of the kinetic and potential energy of the planet in its orbit is a constant, C, given by 1 2 v2 µ r = C, where v is the magnitude of the velocity of the planet when its radius vector is r, µ = G(M + m) and G is the gravitational constant. Show that ( 2 v 2 = µ r 1 ), a where a is the planet s orbital semi-major axis. [11] Halley s Comet moves in an elliptical orbit of eccentricity 0.9673. Calculate the ratio of the magnitudes of (a) its linear velocities, (b) its angular velocities, at perihelion and aphelion. [6]
16 A1X/Y Problems, 2006 2007 3.28e Two planets P 1 and P 2 of masses m 1 and m 2, orbital semi-major axes a 1 and a 2, and orbital periods T 1 and T 2 respectively, are in orbits about a star of mass M. Using the expression ( a 3 ) 1/2 T = 2π, µ where µ = G(M + m) and G is the gravitational constant, derive Newton s form of Kepler s third law. [8] A satellite of Jupiter has an orbital period of 0.498 2 days and an orbital semi-major axis about Jupiter of 0.001 207 AU. Jupiter s orbital period and semi-major axis about the Sun are 11.86 years and 5.203 AU respectively. Calculate the ratio of the mass of Jupiter to that of the Sun. [9] 3.29e Explain the importance of conservations laws for dynamical astronomy calculations, and give two examples of quantities that are conserved during simple gravitational interactions. [5] A body of mass m is in orbit about a planet of mass M( m) taken to be at rest. Write down an expression of the total energy of this system in terms of the body s speed, v, and distance from the centre of the planet, r, and show that ( ) 2 v 2 = GM r + C where C is a constant for the orbit. [4] It has been suggested that liquid oxygen fuel pods, mined from the Moon s South Pole, could be launched to waiting spacecraft in orbit at a radius of 2 lunar radii about the Moon. Given that C = 1/a, where a is the semi-major axis of the orbit, use the above equation to determine the initial velocity required (a) launching the pods vertically from the surface, [3] (b) launching the pods parallel to the surface. [3] What other major factor would be important to complete the transfer? Explain why (b) might be preferred overall. [2] 4 Solar System Physics 4.1e List five main differences between the terrestrial and jovian planets. [5] 4.2e Give a sketch of the Earth s interior, clearly indicating the different regions, and their approximate dimensions. [5] 4.3e Explain the terms (a) plate tectonics (b) volcanism (c) outgassing. [5]
4 Solar System Physics 17 4.4e Give a sketch of the atmosphere of the Sun, indicating the approximate dimensions of the various regions and their approximate temperatures. [5] 4.5e The Earth and Jupiter s satellite Io both exhibit volcanism. Briefly describe the causes of this volcanism in each case. [5] 4.6e Sketch the internal structure of the Sun, indicating the approximate dimensions and temperatures of the various regions. [5] 4.7e Explain the mechanism of the greenhouse effect. Give the names of two main greenhouse gases in the Earth s atmosphere. [5] 4.8e In what way do the surface features of the Moon and Earth radically differ and why? [5] 4.9e The surface temperature of Venus is 700 K and that of the Earth is 300 K. Calculate the wavelengths at which the radiative flux from each planet is maximum. You may use Wien s law, λ max T = 2.9 10 3 m K. [2] In what region of the spectrum do these wavelengths lie? [1] Explain carefully what is meant by the greenhouse effect in planetary atmospheres. Discuss this with reference to the atmospheres of the terrestrial planets. [10] How does the runaway greenhouse effect explain the near absence of water in Venus s atmosphere? [4] 4.10e Explain briefly what evidence there is to support the statement that the Moon s age is about 5 000 million years? [4] The abundance of the radioactive isotope of potassium, 40 19K, found in a sample of rock found on the Moon is found to be 1 part in 10 8, whereas the abundance of argon, 40 18Ar, is found to be 2 parts in 10 9. Assuming that all the argon was produced through the radioactive decay K, estimate the age of the sample of rock and comment on the age you obtain. [8] of 40 19 Explain why the rate ratio of radiogenic heating to the rate of solar heating would be greater in the early life of a planet. Estimate the percentage change in the rate of radiogenic heating due to 40 19K decay over the Moon s lifetime. [5] (You may assume that the number, N, of parent radioactive isotopes obeys the exponential decay law, N = N 0 exp( λt). Take the half-life of 40 19 K to be 1.3 109 years.) 4.11e Show that the surface temperature, T p, of a planet is given by ( ) 1/2 T p = (1 A) 1/4 R T e, 2r where r is the distance of the planet from the Sun, and R is the solar radius, T e is the effective temperature of the Sun, and A is the albedo of the planet. State clearly any assumptions that you make. [10] Assuming that Mars has an albedo of 0.2, calculate its surface temperature given that the orbital period of Mars is 1.9 years. [5]
18 A1X/Y Problems, 2006 2007 What evidence is there for the statement that the surface temperature was considerably higher than this in the past? [2] (For the Sun, T e = 5 800 K.) 4.12 State the main differences between the properties of the terrestrial planets and the jovian planets. [6] Show that the escape velocity is given by v e = (2GM/R) 1/2.(You may assume that the gravitational potential energy of body of mass m on the surface of a sphere of mass M and radius R is given by GMm/R and its kinetic energy is given by mv 2 /2.) Calculate the ratio of the escape velocities for the two planets given the following information: [10] Venus Jupiter Mass in kilograms 4.87 10 24 1.90 10 27 Radius in metres 6.05 10 6 7.14 10 7 How does the difference in escape velocities explain the difference between the chemical composition of the atmospheres of the two planets? {5.85} [4] 4.13 Sketch the monochromatic energy flux, F λ, against wavelength, λ, at the Sun s surface, clearly indicating the wavelength at which the flux is a maximum. You may assume Wien s law. [4] Show that the total radiative flux from the sun falling on a planet s surface is given by ( ) R 2 σ Te 4 r, where r is the distance of the planet from the Sun, R the radius of the Sun, and T e the effective surface temperature of the Sun. Assuming that a planet is in radiative equilibrium, and that a fraction A of the Sun s radiation is reflected by the planetary surface, show that the surface temperature of the planet is given by T p = (1 A) 1/4 ( R 2r ) 1/2 T e. [12] Calculate the surface temperature for Mars given that its albedo, A, is 0.16 and that it is at a mean distance of 1.52 AU from the Sun. {217.2 K} [3] At what wavelength does Mars emit most of its radiation? [3] (You may assume that T e = 5 800 K, and 1 AU= 1.496 10 11 m and that the radius of the Sun is 6.960 10 8 m.) 4.14 If one ignores the effect of Venus s atmosphere the predicted surface temperature would be around 240 K, which is the temperature above the highly reflective cloud. The measured surface temperature however is about 700 K. Explain the reasons for this difference. [6] A simple model of the atmosphere of Venus is described by the figure below.
4 Solar System Physics 19 PSfrag replacements F af F c cloud (1 a)f F V F c r F V surface of Venus F is the radiative flux from the Sun arriving at the top of Venus s atmosphere. The radiative flux from the Sun reflected by the cloud is given by af where a is the albedo, and the radiative flux transmitted through the cloud is (1 a)f. The radiative flux emitted by the upper and lower surface of the cloud is given by F c. A fraction r of the radiative flux, F V, emitted from Venus s surface is reflected by the cloud and the remaining fraction (1 r) is absorbed by the cloud. Assuming the radiative flux is in overall balance, show that F c = (1 a)f and F V = F (1 a) + r F V + F c. Hence show that F V = 2 F c 1 r Assuming that both cloud and Venus s surface radiate as black bodies, and that the cloud is at a temperature of 240 K, calculate the value of r necessary to yield the measured surface temperature of Venus of 720 K. {0.975} [6] 4.15 What is meant by the term greenhouse effect? Explain why it is particularly important for Venus but not for Mars. [5] The surface atmospheric pressure of Mars is 0.006 times that of the Earth. Taking the temperature of the lower atmosphere of Mars to be 250 K calculate the density of its atmosphere. (1 atmosphere is approximately 10 5 N m 2.) Assume for the purposes of this question that Mars atmosphere is 100 % CO 2. The mass number of atomic oxygen, O, is 16 and C is 12. {0.0128 kg m 3 } What is meant by the atmosphere being isothermal? Assuming this to be the case calculate the scale-height of Mars atmosphere, and sketch the atmospheric pressure as a function of height. {12.581 km} At what height is the pressure 0.002 Earth atmospheres? {13.81 km} [2] (Mass of Mars = 6.42 10 23 kg m 3, radius of Mars = 3.39 10 6 m.) 4.16 Explain the terms (a) igneous, (b) sedimentary, (c) metamorphic, and [8] [8] [5]
20 A1X/Y Problems, 2006 2007 (d) primitive rocks. Where might these rocks be found? [8] Calculate the age of a rock sample which contains 46 parts per billion of 40 K and 12 parts per billion of 40 Ar. Assume that all of the 40 Ar found is the direct result of decay from 40 K, with a half-life of 1.3 10 9 years. {4.3 10 8 yr} [8] Explain why this rock would be considered relatively ancient if found on the Earth s surface but not if it were found on the Moon. [4] 4.17 Discuss briefly the internal structure and composition of the terrestrial planets, pointing out the similarities and differences. [8] Explain the term radiogenic heating. [2] The decay of 40 K into 40 Ar is responsible for about 50 % of the radiogenic heating of the Earth today [ 232 Th(orium), 238 U(ranium) and 235 U provide the rest]. 1 kg of Earth contains about 10 8 kg of 40 K. The atomic mass of 40 K is 39.97 amu and the atomic mass of 40 Ar is 39.96 amu. The half life of 40 K is about 109 years. Estimate the amount of heat thus generated per year per kg of Earth material, and hence estimate the power in watts due to radiogenic heating for the whole Earth. { 2 10 4 J yr 1 ; 3 10 13 W} [8] How does this compare with the rate of heating due to the Sun? { 10 17 W} [2] (You may assume that the mass of the Earth is about 6 10 24 kg and the solar constant is 1.4 10 3 W m 2.) 4.18 Explain what is meant by a tidal force and explain in qualitative terms the origin and meaning of the Roche stability limit. [8] The Roche stability limit of a planet is approximately given by 2.5 times the planetary radius. If the radius of Jupiter is 71 000 km and Saturn 60 000 km, estimate the likely maximum distance that a ring could be found from the centre of either planet. {1.8 10 5 km; 1.5 10 5 km} [4] Give a possible explanation for the formation and the structure of these rings. [4] Describe the asteroid belt, and give a plausible explanation for the Kirkwood gaps. [4] 4.19 Outline the currently accepted view of how the solar system formed from interstellar material. Describe the main features of the solar system that this model explains (its structure, chemical composition, age and orbits/motion of planets). [20] [Your answer should be succinct, factual, and about 300 words long (no more than 400). It can be in note form.] 4.20 Discuss what is meant by Bode s Law and briefly describe its significance in relation to a physical understanding of why the planets are found at their respective distances from the Sun. Another expression relating the planetary mean distances, r, from the Sun (in AU) according to their number order, n, is n = 3 + 4.11 log 10 r. Using the tabulated information given in your lecture notes, how well do Venus, Earth and Jupiter fit the equation?
4 Solar System Physics 21 4.21 From the data given in your lecture notes, calculate the average density of each of the planets compared to the Earth. Group the results according to any broad features you observe. 4.22 Show that the tidal force acting on a body, mass m, radius R, when at a distance r from an object of mass M, is given by F tidal = 4GMm R r 3, by considering the difference in gravitational forces exerted on the far and near side of the affected body. Assume r R. Calculate the ratio of the tidal forces exerted by Jupiter on its satellites Io and Callisto. Explain how the difference in tidal forces between these two satellites is reflected in their physical properties. (r Io = 4.22 10 5 km, r Callisto = 1.88 10 6 km, m Io = 8.92 10 22 kg, m Call = 1.08 10 23 kg, D Io = 3630 km, D Callisto = 4 800 km) {55.2} 4.23 Calculate the ratio of the surface gravities of the Moon and Mercury. Explain the consequences of the answer in terms of the impact cratering on each body. (M Moon = 0.012M Earth, M Mercury = 0.055 8M Earth, D Moon = 0.27D Earth, D Mercury = 0.381D Earth ) {0.43} 4.24 Compare the surface gravity of Jupiter and Saturn. By considering the lower atmospheres of each planet, estimate the altitude difference between the formation of NH 3 and H 2 0 clouds for Jupiter and Saturn. Is this consistent with a scale-length h = kt/(mg) for atmospheric variations? Use your notes to get the atmospheric data. (M Jup = 318M Earth, M Sat = 95.2M Earth, D Jup = 10.86D Earth, D Sat = 9D Earth ) 4.25 Calculate the age of a rock sample which contains 46 parts per billion of 40 K and 12 parts per billion of 40 Ar. Assume that all of the 40 Ar found is the direct result of decay from 40 K, with a half-life of 1.3 billion years. Would this rock be considered relatively ancient if found on the Earth s surface? What if it were a Moon rock? {4.35 10 8 yr} 4.26 The angular diameter of the Sun as seen from Earth is 32 arcmin. Calculate the radius of the Sun. Assuming the Sun emitted its luminosity like a perfect black body, calculate its effective temperature, and compare it with Earth s surface temperature. 4.27 In association with the apparent brightness of an asteroid, define the term absolute magnitude. Demonstrate how absolute magnitude depends on the physical size of the body and on its albedo. Show that the apparent magnitude, m, of an asteroid may be expressed by m = m 0 + 5 log 10 r + 5 log 10 D, where m 0 is the absolute magnitude of the asteroid, r its distance from the Sun and D its distance from the Earth. The asteroids Ceres and Vesta have had their diameters measured directly as 1 000 km and 540 km respectively. At opposition their magnitudes are 6.7 and 5.3 and their solar distances 2.77 and 2.36 AU. Calculate the ratio of their albedos. {0.187}
22 A1X/Y Problems, 2006 2007 4.28 A spherical asteroid has a mean absolute magnitude of 10 and is observed at a heliocentric distance of 1.5 AU and at an Earth distance of 1.2 AU. It rotates about an axis perpendicular to its orbital plane which is coplanar to the Earth s orbital plane. During its rotation, it alternately presents two distinct hemispheres, one with an albedo of 0.10, the other with 0.05. Sketch out the light curve (magnitude against time quantitatively) for the object. 4.29 Explain how measurements of the polarization of the light from an asteroid over a range of phase angles can lead to an estimation of its albedo and hence to a determination of its diameter. Two asteroids with orbital radii 2.2 and 2.5 AU have the same apparent magnitude at opposition. The respective values of the polarization at the turning point, P min, on the P(q) curve are 1 % and 0.5 % respectively. Determine the ratio of their radii. 4.30 The equation for hydrostatic equilibrium is dp/dz = gρ where z is the height above the surface, g the surface gravitational acceleration, p the pressure and ρ the gas density. Using the ideal gas law p = ρkt/µ, where T is the temperature, k is Boltzmann s constant and µ the average mass of a gas particle, show that in an isothermal atmosphere p = p 0 exp( z/h), where H = kt 0 /(µg). We think of H as the scale-height for variations in the atmosphere. What could the pressure profile be if the atmosphere was not isothermal? 4.31 Suppose the surface of a sphere the same size as the Sun was covered in electric light bulbs. What would the wattage of each bulb have to be in order to match the Sun s luminosity? Assume the surface area of a bulb is 30 cm 2. 4.32 Compare the tidal force exerted by the Sun on the Earth with that exerted by the Moon on the Earth. Explain your answer in terms of the spring and neap tides. 4.33 The Stefan-Boltzmann Law is L = 4π R 2 σ T 4, and Wien s Law is λ = W/T. Describe how these formulae can be applied to planetary energy budgets, explaining what each symbol means. Mars has an orbital radius about the Sun of a = 1.524 AU, and a planetary radius of 3.38 10 3 km. Ignoring Mars s atmosphere, calculate: (a) the power supplied by the Sun to Mars, (b) the power radiated by Mars assuming the planet has an albedo of 0.15, (c) the dominant wavelength of such emissions from Mars. Given that observations show that Mars radiates predominantly at a wavelength of 1.45 10 5 m, compare this with your answer for the dominant wavelength, and account for any discrepancy. (W = 2.90 10 3 km) 4.34 State Wien s Law, and describe briefly how it may be used. What is the effective temperature of an object? Radiation from Mars is observed predominantly at a wavelength of λ = 1.45 10 5 m. Calculate the surface temperature and hence the implied luminosity of Mars. The effective temperature of Mars is 217 K. Calculate the intrinsic luminosity of Mars using this temperature, and compare it with your earlier answer. Can you explain the difference? Given that Mars has a mean orbital radius a = 2.3 10 11 m, calculate its albedo. (Planetary radius of Mars R M = 3.4 10 6 m, W = 2.9 10 3 K m)